Our program begins in two dimensions and relies crucially on the fact that the Schur index of any 4d \({\cal N}=2\)  superconformal field theory can be interpreted as the vacuum character of a 2d VOA. As mentioned in the introduction, for a generic VOA there is no reason to expect that the vacuum character should have simple modular properties. We will take as a given the central conjecture of [1]}, which equates the Higgs branch of the 4d \({\cal N}=2\)  theory with the associated variety of the corresponding VOA. A corollary of this conjecture is that any VOA with a 4d avatar is “quasi-lisse”,We remind the reader that quasi-lisse VOAs [2]} are those whose associated varieties are symplectic, or more generally symplectic singularities [3]}, [4]}, [5]} with a finite number of symplectic leaves. and as such its vacuum character must satisfy a monic modular differential equation (MDE) [2]}.Explicit closed form expressions for Schur indices in terms of quasi-modular forms were recently obtained in [7]}, [8]} for a large class of SCFTs.
